Of a group of seven women, Mary is one of them, and of four men, John is one of them. How many commissions can be formed with any number of people, provided there are the same number of men and women?
I did so:
4 men and 4 women: $${7 \choose 4} { {4} \choose {4} } = 35 $$
3 men and 3 women:
$${7 \choose 3} { {4} \choose {3} } = 140 $$
2 men and 2 women:
$${7 \choose 2} { {4} \choose {2} } = 126 $$
1 man and 1 woman:
$${6 \choose 1} { {4} \choose {1} } = 24 $$
By the additive principle, it comes: $$35 + 140 + 126 + 24 = 325$$
The answer is 329. Have I forgotten a case or addressed the problem in the wrong way?